Overview
In the previous sessions, we studied the Communication, Problem Solving, and Reasoning and Proof Standards. In this session, we examine the Representation Standard. Representation includes the ways that students depict their mathematical thinking as well the process they use to put their thinking into that form. Representations can include a variety of written formats, oral explanations, models with manipulative materials, or even the mental process one uses to do mathematics. While representation is the focus of this session, it is helpful to keep in mind that it often works in conjunction with other process standards; for instance, representation can be key to effective problem solving and communication.
We want students to represent their mathematical thinking for two purposes: so that they better understand the mathematics, and so that they can share their ideas with others. Some forms of representation are diagrams, graphical displays, and symbolic expressions. However, before students are ready to use these conventional forms, they need opportunities to express their thinking using their own invented, non-conventional forms of representation. In this session, we look at both non-conventional and conventional forms that students in the middle grades might use to represent their mathematical thinking.
Learning Objectives
This session shows how to help students do the following:
Create representations from given data
Use representations to solve problems
Use representations to model mathematical problems
Translate among representations
Use representations to communicate mathematical knowledge
NCTM Representation Standard
Instructional programs. . . should enable all students to --
Create and use representations to organize, record, and communicate mathematical ideas
Select, apply, and translate among mathematical representations to solve problems
Use representations to model and interpret physical, social, and mathematical phenomena
Principles and Standards of School Mathematics (NCTM, 2000, p. 67). For more information on this process standard, see the NCTM Web site.
Defining Representation
When we solve mathematical problems, a core part of the solution process is how we represent the ideas in the problem. The form of representation we select allows us to manipulate the information (as opposed to manipulating symbols) to reach a sensible solution. With the Representation Standard, it is our goal for students to learn to recognize, compare, and use an array of representational forms for the concepts they are learning in the middle grades.
Representation is not about manipulating symbols. Rather, it requires students to generate or select a meaningful way to show the facts in a mathematical situation. Using representation helps them identify patterns and relationships and make predictions.
Students in the middle grades should use multiple representations, choosing the appropriate representation for a particular problem situation. Examples of forms of representation include models, drawings, tables, graphs, spreadsheets, expressions, and equations. When students gain access to mathematical representations and the ideas they represent, they have a set of tools that enable them to organize their thinking and increase their ability to think mathematically.
It is not usually apparent which form of representation to use when students first approach a problem. They might find that their own representation is the most comfortable with which to begin. Part of problem solving is trying, and sometimes rejecting, possible forms of representation when choosing an effective one to use.
In the Building Viewpoints video from the Observe part of this session, the students are given a two-dimensional description of a three-dimensional structure. From that description, they build the structure and then closely examine each view. In order to communicate their thinking, they represent each view on grid paper. From that information, they can determine relationships among the views and describe the original structure. The culmination of this activity is learning how to represent three-dimensional objects in a diagram. For example, students can create two-dimensional drawings on grid paper that represent the three-dimensional aspects of their buildings.
In the middle grades, we want students to use representations to solve problems, to clarify and extend their mathematical ideas, and to recognize that two different representations might describe the same phenomenon.
Drawings and models are especially appropriate as students work with geometry and spatial visualization concepts. However, these representations can also connect what students know in one strand to their experiences in other strands. For example, students who use area models to develop an understanding of multiplication can connect the concept of multiplication of whole numbers to finding the area of two-dimensional figures. Fraction area models help students develop understanding of proportional reasoning. Area models can also be used to develop concepts in probability. This single form of representation, first used with a simple concept, becomes a tool for students to use to help give meaning to more complex ideas.
In addition to visual representations, students can organize information by making organized lists and tables. From the lists, they can find, describe, and extend patterns and make generalizations for a variety of cases. Their tables can be transformed into visual representations by drawing graphs. The information in the graph then shows the relationships between the elements in the table. If an element increases at a constant rate, the graph will be linear. If an element increases exponentially, the graphic representation will be much different.
It is important for students to use representations that are meaningful. At first, students may choose to put data into a bar graph, although, as we saw in the Interpreting Graphs and Stories problem, a line graph might actually be more appropriate. However, if students can gain understanding from a bar graph, they later can extend their thinking to other graphic representations and transfer that meaning to those graphs as well.
Why do many students in the middle grades start with bar graphs? Mathematics in the elementary grades gives students many opportunities to collect data and represent it in bar graphs. For this reason, students are comfortable with this representation. In the middle grades, they will be given a variety of opportunities to extend their experience with bar graphs to linear graphs. As algebraic concepts become an integral part of the middle-grades curriculum, the transition of understanding from simple to more complex graphs will have meaning for students if we build on their earlier experiences. Remember, we want to use representations to support students in developing a deeper understanding of the mathematical concepts -- not as an end to themselves.
The Role of the Teacher
The teacher's role in setting the classroom environment for the Representation Standard is similar to the other process standards. Selecting rich mathematical tasks that utilize a variety of representations is the first step. As students work to complete the task, they communicate with one another to select appropriate representations that will help them see relationships. The teacher poses questions that help students move from non-conventional to conventional representations and interpret the mathematical ideas that develop from those representations.
The Role of Technology
An additional consideration in working with various representations in the middle grades is the role of technology. Students who are developing tables and seeing patterns can apply that knowledge to a spreadsheet. As students transfer their work from their own table to a spreadsheet, they begin to think about expressions that will enable them to utilize the technology to extend their tables and generalize patterns. The use of graphing calculators also enables students to explore how changing one variable in a problem affects the outcome. Dynamic geometry software gives students opportunities to explore geometric properties with graphic representations on the computer. Technology offers multiple opportunities for students to represent and develop mathematical concepts in the middle grades.
An additional resource is the World Wide Web. Interactive activities, such as the "e-examples" on the NCTM Web site, use technology to represent mathematical ideas in an accessible, hands-on way. These activities can also be considered for use in your classroom. For additional information about technology, see the Reference Section.
In the previous sessions, we studied the Communication, Problem Solving, and Reasoning and Proof Standards. In this session, we examine the Representation Standard. Representation includes the ways that students depict their mathematical thinking as well the process they use to put their thinking into that form. Representations can include a variety of written formats, oral explanations, models with manipulative materials, or even the mental process one uses to do mathematics. While representation is the focus of this session, it is helpful to keep in mind that it often works in conjunction with other process standards; for instance, representation can be key to effective problem solving and communication.
We want students to represent their mathematical thinking for two purposes: so that they better understand the mathematics, and so that they can share their ideas with others. Some forms of representation are diagrams, graphical displays, and symbolic expressions. However, before students are ready to use these conventional forms, they need opportunities to express their thinking using their own invented, non-conventional forms of representation. In this session, we look at both non-conventional and conventional forms that students in the middle grades might use to represent their mathematical thinking.
Learning Objectives
This session shows how to help students do the following:
NCTM Representation Standard
Instructional programs. . . should enable all students to --
Principles and Standards of School Mathematics (NCTM, 2000, p. 67). For more information on this process standard, see the NCTM Web site.
Defining Representation
When we solve mathematical problems, a core part of the solution process is how we represent the ideas in the problem. The form of representation we select allows us to manipulate the information (as opposed to manipulating symbols) to reach a sensible solution. With the Representation Standard, it is our goal for students to learn to recognize, compare, and use an array of representational forms for the concepts they are learning in the middle grades.
Representation is not about manipulating symbols. Rather, it requires students to generate or select a meaningful way to show the facts in a mathematical situation. Using representation helps them identify patterns and relationships and make predictions.
Students in the middle grades should use multiple representations, choosing the appropriate representation for a particular problem situation. Examples of forms of representation include models, drawings, tables, graphs, spreadsheets, expressions, and equations. When students gain access to mathematical representations and the ideas they represent, they have a set of tools that enable them to organize their thinking and increase their ability to think mathematically.
It is not usually apparent which form of representation to use when students first approach a problem. They might find that their own representation is the most comfortable with which to begin. Part of problem solving is trying, and sometimes rejecting, possible forms of representation when choosing an effective one to use.
In the Building Viewpoints video from the Observe part of this session, the students are given a two-dimensional description of a three-dimensional structure. From that description, they build the structure and then closely examine each view. In order to communicate their thinking, they represent each view on grid paper. From that information, they can determine relationships among the views and describe the original structure. The culmination of this activity is learning how to represent three-dimensional objects in a diagram. For example, students can create two-dimensional drawings on grid paper that represent the three-dimensional aspects of their buildings.
In the middle grades, we want students to use representations to solve problems, to clarify and extend their mathematical ideas, and to recognize that two different representations might describe the same phenomenon.
Drawings and models are especially appropriate as students work with geometry and spatial visualization concepts. However, these representations can also connect what students know in one strand to their experiences in other strands. For example, students who use area models to develop an understanding of multiplication can connect the concept of multiplication of whole numbers to finding the area of two-dimensional figures. Fraction area models help students develop understanding of proportional reasoning. Area models can also be used to develop concepts in probability. This single form of representation, first used with a simple concept, becomes a tool for students to use to help give meaning to more complex ideas.
In addition to visual representations, students can organize information by making organized lists and tables. From the lists, they can find, describe, and extend patterns and make generalizations for a variety of cases. Their tables can be transformed into visual representations by drawing graphs. The information in the graph then shows the relationships between the elements in the table. If an element increases at a constant rate, the graph will be linear. If an element increases exponentially, the graphic representation will be much different.
It is important for students to use representations that are meaningful. At first, students may choose to put data into a bar graph, although, as we saw in the Interpreting Graphs and Stories problem, a line graph might actually be more appropriate. However, if students can gain understanding from a bar graph, they later can extend their thinking to other graphic representations and transfer that meaning to those graphs as well.
Why do many students in the middle grades start with bar graphs? Mathematics in the elementary grades gives students many opportunities to collect data and represent it in bar graphs. For this reason, students are comfortable with this representation. In the middle grades, they will be given a variety of opportunities to extend their experience with bar graphs to linear graphs. As algebraic concepts become an integral part of the middle-grades curriculum, the transition of understanding from simple to more complex graphs will have meaning for students if we build on their earlier experiences. Remember, we want to use representations to support students in developing a deeper understanding of the mathematical concepts -- not as an end to themselves.
The Role of the Teacher
The teacher's role in setting the classroom environment for the Representation Standard is similar to the other process standards. Selecting rich mathematical tasks that utilize a variety of representations is the first step. As students work to complete the task, they communicate with one another to select appropriate representations that will help them see relationships. The teacher poses questions that help students move from non-conventional to conventional representations and interpret the mathematical ideas that develop from those representations.
The Role of Technology
An additional consideration in working with various representations in the middle grades is the role of technology. Students who are developing tables and seeing patterns can apply that knowledge to a spreadsheet. As students transfer their work from their own table to a spreadsheet, they begin to think about expressions that will enable them to utilize the technology to extend their tables and generalize patterns. The use of graphing calculators also enables students to explore how changing one variable in a problem affects the outcome. Dynamic geometry software gives students opportunities to explore geometric properties with graphic representations on the computer. Technology offers multiple opportunities for students to represent and develop mathematical concepts in the middle grades.
An additional resource is the World Wide Web. Interactive activities, such as the "e-examples" on the NCTM Web site, use technology to represent mathematical ideas in an accessible, hands-on way. These activities can also be considered for use in your classroom. For additional information about technology, see the Reference Section.